Bases in Lie and Quantum Algebras

TL;DR

This paper introduces a unique basis for quantum algebras derived from Lie bialgebra structures, providing a canonical way to classify quantum groups and clarifying the relationship between Lie and quantum algebra bases.

Contribution

It establishes a unique, canonical basis for quantum algebras based on the Drinfeld double structure, advancing the classification of quantum groups.

Findings

The analytical basis is uniquely determined for each quantum group.

The construction links quantum algebra classification to bialgebra classification.

Explicit examples for su(2) and su_q(2) illustrate the method.

Abstract

Applications of algebras in physics are related to the connection of measurable observables to relevant elements of the algebras, usually the generators. However, in the determination of the generators in Lie algebras there is place for some arbitrary conventions. The situation is much more involved in the context of quantum algebras, where inside the quantum universal enveloping algebra, we have not enough primitive elements that allow for a privileged set of generators and all basic sets are equivalent. In this paper we discuss how the Drinfeld double structure underlying every simple Lie bialgebra characterizes uniquely a particular basis without any freedom, completing the Cartan program on simple algebras. By means of a perturbative construction, a distinguished deformed basis (we call it the analytical basis) is obtained for every quantum group as the analytical prolongation of the above defined Lie basis of the corresponding Lie bialgebra. It turns out that the whole construction is unique, so to each quantum universal enveloping algebra is associated one and only one bialgebra. In this way the problem of the classification of quantum algebras is moved to the classification of bialgebras. In order to make this procedure more clear, we discuss in detail the simple cases of su(2) and su_q(2).